When I first started playing kickball I did not have the energy to run from first to home or the power to throw the ball all the way from home to 1st. Here are the workouts for powering up specific movements in kickball with various workouts.
Standard disclaimer, I am not a doctor, consult one before doing any new physical activity.
How to kick the ball further
When you wind up to kick the big red ball you are primarily using the muscles of your upper leg, mainly the hamstrings, quadriceps and glutes. The upper leg is what drives the power but you are also using core, hip and foot muscles for stabilization and control. Lucky for us, the barbell squat (low bar and front) hits pretty much all these muscles. My preference for learning and training the squat is what is done in Starting Strength. I would recommend picking up the book if you are serious about weight training, but the website wiki on how to do and train the squat is good enough to get started.
How to run the bases faster
Want to beat the throw to 1st base? Going to need to move faster to do that. My favorite exercise for this, and something that allowed me to shave off 50% from my base time was doing running intervals. To go faster you will need to run faster and this is what intervals will help with. I have seen distance runners that were awful at running bases. Their mistake is thinking being able to do 13.1 miles at 9 minutes per mile translates to being able to run 50 feet at a 4 mile pace.
Intervals are super simple and quick to do. A complete interval workout should only take at most 20 minutes (including warmup and cooldown) and will leave you exhausted. You will need to have something to count down from 30 seconds. I actually just count down from 30 seconds in my head but an interval app would be more ideal Do a slow warm up jog for 5 minutes. Once warmed up, you will start doing your intervals. For 30 seconds you run top out, as absolutely fast as you can go and make sure to not hold anything back. At the end of 30 seconds you can either switch to a slow jog or even a walk. After 60 seconds of walking / jogging do another full out 30 second run. Repeat this interval for 10 minutes and end with a 5 minute cool down job. Do this run no more than once a week. Each week, try to travel a further distance / faster speed in each 30 second split.
If this is not a workout you have done before, be ready to have some immediate gains in your base running speed as well as your ability to not get as tired when you have to do multiple runs in a row (stupid kicker keeps on fouling!).
How to throw further
Caught a ball in the outfield and you need to get the ball to home base? Trying to throw the ball faster to 1st base to get out the runner? You are going to need to strengthen up your arms, shoulders and core to do this.
There are a couple of good workouts for this, mostly because there are several different ways to throw a ball. For a good base of strength, start with the push up. This is a great overall chest, arm and core workout that will strengthen all the muscles you need to use for throwing a ball hard and long. The greatest benefit of the push up is you can do this anywhere, so, no excuses!
Once you can do more than 20 pushups at one time, you are no longer working on strength and you are now working on endurance. To continue adding strength to your chest and arms, it is time to switch over to the bench press. Just like the squat, I recommend Starting Strength for getting into the bench press. The Starting Strength wiki has a good page for training and learning how to bench to get you started.
In addition to the bench press for a base of strength, the second exercise I would suggest would be the medicine ball pass. This is a good drill you can work into your kickball practices since you will need another person. Get yourself a 5 or 10 lb medicine ball, I would recommend this one, get yourself a partner and then, carefully, throw the ball back and forth. Alternate between different throws focusing on chest passes.
Practice some of your harder throws without using a partner, since you do not want to injure the other person. This requires some back and forth but is worth it. As you get stronger, use a heavier and heavier ball.